\section{Descriptive Statistics}
Descriptive statistics is useful for determinig characteristical values from a
sample. These values are:
\begin{table}[htb]
	\centering
	\listengroesse
	\begin{tabular}{l|c|c}
    \textbf{Characteristic Value} & \textbf{Formula} & \textbf{Excel Formula} \\
    \hline
    \hline
	Sample size & $ \text{number of values} $ & count(A1:A20) \\
	\hline
	Average & $ \overline{x} = \frac{1}{n} \sum x_i $ &
	average(A1:A20) \\
	\hline 
	Variance & $ \frac{1}{n-1} \sum(x_i-x)^2 $ & var(A1:A20) \\
	\hline 
	Standard deviation & $ \sqrt{Variance} $ & stdev(A1:A20) \\
	\hline
	Minimum & $ \text{Mimumum of sample} $ & min(A1:A20) \\
	\hline
	Maximum & $ \text{Maximum of sample} $ & max(A1:A20) \\
	\hline
	Quartiles & $ \text{25\%, 50\% or 75\% Quartile} $ & quartile(A1:A20, 1) (25\%)
	\\
	\hline
	Quantiles & $ \text{1\% - 99\% quantiles} $ & percentile(A1:A20, 0.23) (23\%)
	\\
	\hline
	Geometric mean & $ x_g = \sqrt[n]{\prod x_n} $ & geomean(A1:20) \\
	\hline
	Median & $ x_{0.5} = x_{\left[ \frac{n+1}{2} \right]} $ & median(A1:A20) \\
	\hline
	Robust Mean & $ x_r = \overline{x} \text{ without upper and lower 5\%} $ &
	by hand \\
	\hline
	Range & $ max - min $ & max(A1:A20) - min(A1:A20) \\
	\hline
	Coefficient of variation & $ vk = \frac{s}{x} $ & $
	\frac{stdev(A1:A20)}{A1:A20} $
	\\
	\hline
	Quartile ratio & $ \frac{x_{0.25}}{x_{0.75}} $ & $ \frac{quartile(A1:A20,
	1)}{quartile(A1:A20, 3)} $ \\
	\hline
	Min/Max ratio & $ \frac{min}{max} $ & $ \frac{min(A1:A20)}{max(A1:A20)}
	$ \\
	\hline
	Skewness & $ \overline{x} - x_{0.50} $ & mean(A1:A20) - median(A1:A20)
    \end{tabular} 
\end{table}
